Remark 7.6.2.6 (Powers as Limits). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$. Then a morphism of simplicial sets $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ can be identified with a natural transformation $\alpha : \underline{X} \rightarrow \underline{Y}$, where $\underline{X}, \underline{Y}: K \rightarrow \operatorname{\mathcal{C}}$ denote the constant diagrams taking the values $X$ and $Y$, respectively. In this case:
The natural transformation $\alpha $ exhibits the object $X$ as a limit of the diagram $\underline{Y}$ (in the sense of Definition 7.1.1.1) if and only if $e$ exhibits $X$ as a power of $Y$ by $K$ (in the sense of Definition 7.6.2.1).
The natural transformation $\alpha $ exhibits the object $Y$ as a colimit of the diagram $\underline{X}$ (in the sense of Definition 7.1.1.1) if and only if $e$ exhibits $Y$ as a tensor product of $X$ by $K$ (in the sense of Definition 7.6.2.1).